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Mathematics > Group Theory

Title:Exponential growth rates of free and amalgamated products

Abstract: We prove that there is a gap between $\sqrt{2}$ and $(1+\sqrt{5})/2$ for the
exponential growth rate of free products $G=A*B$ not isomorphic to the infinite
dihedral group. For amalgamated products $G=A*_C B$ with
$([A:C]-1)([B:C]-1)\geq2$, we show that lower exponential growth rate than
$\sqrt{2}$ can be achieved by proving that the exponential growth rate of the
amalgamated product $\mathrm{PGL}(2,\mathbb{Z})\cong (C_2\times C_2) *_{C_2}
D_6$ is equal to the unique positive root of the polynomial $z^3-z-1$. This
answers two questions by Avinoam Mann [The growth of free products, Journal of
Algebra 326, no. 1 (2011) 208--217].